How to find out the tier of a university in the USA?

Typically, when people talk about tiers in context of universities, the assumption is that they refer to levels of teaching and/or research activity within some standardized framework. As far as I know, the most well-known framework of this kind is Carnegie Classification of Institutions of Higher Education. Based on this question, I further assume that it implies an interest in university research tiers (which matches the traditional Carnegie Classification framework - the Basic criterion).

Thus, in order to find a research tier of a university, click LOOKUP link in the main menu of the above-mentioned website and enter your search criteria on the Institutional Lookup page. The request will generate a list of results or a single result, depending on your criteria. Click on the linked title of the relevant institution and browse the resulting webpage. On that page, find Basic classification row, which contains the target value. For example, performing the search for my current institution (employer), Georgia Institute of Technology (aka Georgia Tech), we find that it belongs to the category of Doctoral Universities: Highest Research Activity. This is what usually refers to as (the highest) R1 tier (for more details about the shorthand labels, see this page).

Note that, while level of research activity is the most popular classification criterion, there are other criteria (see Listings -> Standard Listings). Also note that Carnegie Classification is focused on the academic ecosystem in the United States. I am not familiar with similar national or international frameworks (but have no doubt that some exist).

I don't know if it includes a "tier" designation anywhere, but the Times Higher Education website has published rankings and there is a very handy search tool where you can sort universities based on the calibre of research they were evaluated to have. Another page discusses the methodology used to arrive at these rankings, and Wikipedia also lists criteria used. This isn't a perfect list of Tiers, but it should be a good place to start when searching for universities conducting quality research.